MinT - Best Known (172, 172+33, s)-Nets in Base 3

Best Known (172, 172+33, s)-Nets in Base 3

(172, 172+33, 1480)-Net over F₃ — Constructive and digital

Digital (172, 205, 1480)-net over F₃, using

3 times m-reduction [i] based on digital (172, 208, 1480)-net over F₃, using
- trace code for nets [i] based on digital (16, 52, 370)-net over F₈₁, using
  - net from sequence [i] based on digital (16, 369)-sequence over F₈₁, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 16 and N(F) ≥ 370, using
      - a function field by GarcÃa and Quoos [i]

(172, 172+33, 9858)-Net over F₃ — Digital

Digital (172, 205, 9858)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(3²⁰⁵, 9858, F₃, 2, 33) (dual of [(9858, 2), 19511, 34]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3²⁰⁵, 19716, F₃, 33) (dual of [19716, 19511, 34]-code), using
  - constructionÂ X applied to C_e(33) âŠ‚ C_e(28) [i] based on
    1. linear OA(3¹⁹⁹, 19683, F₃, 34) (dual of [19683, 19484, 35]-code), using an extension C_e(33) of the primitive narrow-sense BCH-code C(I) with length 19682Â = 3⁹âˆ’1, defining interval IÂ =Â [1,33], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 34 [i]
    2. linear OA(3¹⁷², 19683, F₃, 29) (dual of [19683, 19511, 30]-code), using an extension C_e(28) of the primitive narrow-sense BCH-code C(I) with length 19682Â = 3⁹âˆ’1, defining interval IÂ =Â [1,28], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 29 [i]
    3. linear OA(3⁶, 33, F₃, 3) (dual of [33, 27, 4]-code or 33-cap in PG(5,3)), using
      - discarding factors / shortening the dual code based on linear OA(3⁶, 48, F₃, 3) (dual of [48, 42, 4]-code or 48-cap in PG(5,3)), using
        large caps in PG(5,Â b) [i]

(172, 172+33, 4119107)-Net in Base 3 — Upper bound on s

There is no (172, 205, 4119108)-net in base 3, because

1 times m-reduction [i] would yield (172, 204, 4119108)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 21 514801 732738 771496 736585 689358 148774 643905 018356 386956 661627 492810 698834 956651 613873 194534 294657 > 3²⁰⁴ [i]